a cylinder with a top and bottom has a radius 2x+1 and height 2x-1 . Write a simplifed expression for its

a) surface area, SA= 2pie^2 + 2pierh

b) volume, where V=pier^2h

Hi! Sorry I'm pretty sure that for r and h in the equation for SA there is multiplication instead of addition... also in the Volume equation when there are brackets, why did you add everything instead of adding in the brackets separately and then multiplying the result?

r = radius

h = height

A = Area

V = Volume

With the top and bottom, the surface area is:

A = 2 r ^ 2 pi + 2 r pi h

Volume is equal to the multiple of the area of the base and the height or V=(base area)(height).

V = r ^ 2 pi h

In this case:

r = 2 x + 1

h = 2 x - 1

A = 2 r ^ 2 pi + 2 r pi h =

2 pi r ( r + h ) =

2 pi ( 2 x + 1 ) ( 2x + 1 + 2 x - 1 ) =

2 pi ( 2 x + 1 ) 4 x =

8 pi ( 2 x + 1 ) x =

8 pi ( 2 x ^ 2 + x )

V = r ^ 2 pi h

V = ( 2 x + 1 ) ^ 2 pi ( 2 x - 1 ) =

pi ( 4 x ^ 2 + 4 x + 1 ) ( 2 x - 1 ) =

pi ( 4 x ^ 2 2 x + 4 x 2 x + 2 x - 4 x ^ 2 - 4 x - 1 ) =

pi ( 8 x ^ 3 + 4 x ^ 2 - 2 x - 1 )

a) Well, when life gives you a cylinder, it's time to put your thinking cap on! The surface area (SA) of this witty cylinder can be simplified using the formula SA = 2πr^2 + 2πrh. So all you need to do is substitute the given expressions for the radius (2x+1) and height (2x-1) into the formula. Get your math game on, my friend!

b) When it comes to volume, this cylinder doesn't like to be left hanging either! The volume (V) of the cylinder can be determined using the formula V = πr^2h. Just like before, plug in the given expressions for the radius (2x+1) and height (2x-1) into the formula. Don't forget to embrace the mathematical beauty of simplification!

To find the surface area of the cylinder, we can use the formula SA = 2πr^2 + 2πrh, where r is the radius and h is the height.

a) Surface area (SA):
Given that the radius of the cylinder is 2x + 1 and the height is 2x - 1, we can substitute these values into the formula:

SA = 2π(2x + 1)^2 + 2π(2x + 1)(2x - 1)
= 2π(4x^2 + 4x + 1) + 2π(4x^2 - 1)
= 8πx^2 + 8πx + 2π + 8πx^2 - 2π
= 16πx^2 + 16πx

Therefore, the simplified expression for the surface area of the cylinder is SA = 16πx^2 + 16πx.

To find the volume of the cylinder, we can use the formula V = πr^2h.

b) Volume (V):
Using the same values for the radius and height:

V = π(2x + 1)^2(2x - 1)
= π(4x^2 + 4x + 1)(2x - 1)
= π(8x^3 - 4x^2 + 8x^2 - 4x + 2x - 1)
= π(8x^3 + 4x^2 - 2x - 1)

Thus, the simplified expression for the volume of the cylinder is V = π(8x^3 + 4x^2 - 2x - 1).